Method for predicting creep damage and deformation evolution behavior with time

ABSTRACT

Disclosed is a method for predicting creep damage and deformation evolution behavior with time, which comprises the following steps: obtaining tensile strength σb through high-temperature tensile test; obtaining the strain curve, minimum creep rate {dot over (ε)}m and life tƒ through creep test; obtaining the threshold stress σth at different temperatures; establishing the relationship between the tensile strength σb, the threshold stress σth and the temperature T; establishing the prediction formulas of the minimum creep rate σth and creep life σb based on the threshold stress {dot over (ε)}m and the tensile strength tƒ; establishing a creep damage constitutive model, including strain rate formula and damage rate formula; obtaining the evolution behavior of strain and deformation with time; obtaining the evolution behavior of damage with time.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT/CN2021/141551, filed on Dec.27, 2021 and claims priority of Chinese Patent Application No.202111526072.6, filed on Dec. 14, 2021, the entire contents of which areincorporated herein by reference.

TECHNICAL FIELD

The application relates to a method for predicting creep damage anddeformation evolution behavior with time, and in particular to a methodfor predicting creep damage and deformation evolution behavior with timeby using a creep damage constitutive model.

BACKGROUND

When working under high temperature environment for a long time,high-temperature components are prone to creep deformation, accompaniedby creep damage. Creep damage includes voids, cracks, coarsening ofprecipitates, phase transformation of strengthening phase, oxidation andcorrosion. However, due to the complexity and variety of creep damageforms, it is difficult to quantify the creep damage, characterize thedamage in the process of creep continuously, and describe the evolutionbehavior of creep deformation with time. In addition, the existing creepmodels often describe a single curve, and the fitting parameters have astrong stress-temperature correlation, which is not clearly defined, soit is difficult to achieve reliable extrapolation. Therefore, it isnecessary to develop a prediction method to predict the creepdeformation and the damage evolution behavior in the process ofdeformation, so as to quantitatively evaluate the damage ofhigh-temperature components.

SUMMARY

The objective of the application is to provide a method for predictingcreep damage and deformation evolution behavior with time, so as torealize quantitative evaluation of damage of high-temperaturecomponents.

To achieve the above objective, the application adopts the followingtechnical scheme.

S1, carrying out high-temperature tensile tests of materials atdifferent temperatures T to obtain tensile strength σ_(b) at differenttemperatures;

S2, carrying out high-temperature creep tests under different stressconditions at different temperatures to obtain corresponding creepstrain curves, a minimum creep rate {dot over (ε)}_(m) and a creep lifet_(θ);

S3, obtaining threshold stresses σ_(th) corresponding to differenttemperatures according to the minimum creep rate {dot over (ε)}_(m)obtained in the S2;

S4, establishing a functional relationship between the tensile strengthσ_(b), the threshold stress σ_(th) and the temperature T according tothe tensile strength σ_(b) at different temperatures obtained in the S1and the threshold stresses σ_(th) at different temperatures obtained inthe S3;

S5, on the basis of the threshold stress σ_(th) obtained in the S3 andthe tensile strength σ_(b) obtained in the S1, establishing predictionformulas of the minimum creep rate {dot over (ε)}_(m) and the creep lifet_(ƒ) based on the threshold stress σ_(th) and the tensile strengthσ_(b), respectively, and the minimum creep rate {dot over (ε)}_(m) andthe creep life t_(ƒ) under any stress temperature condition arepredicted by the prediction formulas;

S6, establishing a creep damage constitutive model based on theprediction formulas of the minimum creep rate {dot over (ε)}_(m) and thecreep life t_(ƒ) established in the S5, wherein the creep damageconstitutive model includes a strain rate formula and a damage rateformula;

Step 7, determining parameters in the creep damage constitutive modelestablished in the S6; and

S8, obtaining an evolution behavior of strain deformation with time bysolving the strain rate formula; and obtaining an evolution behavior ofdamage with time by solving the damage rate formula.

In the S3, a relationship between the minimum creep rate {dot over(ε)}_(m), the stress σ and the threshold stress σ_(th) at a σ_(b) sametemperature is established by using the formula {dot over(ε)}_(m)=A_(m)(σ−σ_(th))⁵ according to the minimum creep rate {dot over(ε)}_(m) data obtained from the high-temperature creep tests in the S2,where A_(m) is a constant; doing a same operation for differenttemperatures, and then obtaining threshold stress levels correspondingto different temperatures.

In the S4, a functional relationship between the tensile strength σ_(b),the threshold stress σ_(th) and the temperature T is establishedaccording to the tensile strength σ_(b) at different temperaturesobtained in the S1 and the threshold stresses σ_(th) at differenttemperatures obtained in the S3, a polynomial is used for fitting,

${\sigma_{b} = {\sum\limits_{i = 0}^{n}{a_{i}(T)}^{i}}},{\sigma_{th} = {\sum\limits_{i = 0}^{n}{b_{i}\left( T^{i} \right)}}},$

where n is a number of polynomial terms and a_(i), b_(i) are fittingparameters, i=0,1,2 . . . ,n, ^(n≤3).

In the S5, on the basis of the threshold stress σ_(th) obtained in theS3 and the tensile strength σ_(b) obtained in the S1, the predictionformulas of the minimum creep rate {dot over (ε)}_(m) and the creep lifet_(ƒ) based on the threshold stress σ_(th) and the tensile strengthσ_(b) are respectively established:

${{\overset{.}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}}{{t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}},}$

where A₁, A₂, n₁ and n₂ are constants, σ_(th) is the threshold stress,σ_(b) is the tensile strength, σ is applied stress, T is appliedtemperature, R is a gas constant(R=8.314J/(mo

)), and Q*_(N) is apparent activation energy;

The minimum creep rate {dot over (ε)}_(m) and creep life t_(ƒ) may bepredicted by the above two expressions under arbitrary stresstemperature conditions.

In the S5, the apparent activation energy Q*_(N) is obtained by thefollowing method: under a same

$\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$

value, the apparent activation energy Q*_(N) is determined by a linearfitting straight line slope between a logarithm 1n {dot over (ε)}_(m) ofthe minimum creep rate and the reciprocal 1/T of the temperature.

In the S6, establishing a creep damage constitutive model based on theprediction formulas of the minimum creep rate {dot over (ε)}_(m) and thecreep life t_(ƒ) in the S5:

${\overset{.}{\varepsilon} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}{\exp\left( {\lambda\omega}^{3/2} \right)}}}{{\overset{.}{\omega} = {{\left( \frac{1 - e^{- q}}{q} \right)\left\lbrack \left( {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}} \right. \right\rbrack}^{- 1}{\exp\left( {q\omega} \right)}}},}$

where {dot over (ε)} is the strain rate, {dot over (ω)} is the damagerate, ε is the strain and ω is the damage, q is a constant related tothe temperature, λ is a constant related to the temperature and thestress; in order to ensure that when creep fracture occurs, the damageis 1, λ is defined as a logarithm of the creep rate {dot over(ε)}_(final) to the minimum creep rate {dot over (ε)}_(m) when creepfracture occurs, λ=1n({dot over (ε)}_(final)/{dot over (ε)}_(m));fitting the experimental data, and the expression of λ is established asλ=(α₁T+α₂)σ+(α₃T+α₄), wherein α₁, α₂, α₃ and α₄ are the fittingparameters.

In the S7, the damage rate formula in step 6 is integrated, obtaining:

${\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}},$

where

${t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}}},$

the above-mentioned damage w obtained by an integral is called ananalytical damage;

mathematically transform the strain rate formula in the S6 as follows:

${\omega = \left\lbrack {\frac{1}{\lambda}{\ln\left( {\overset{.}{\varepsilon}/{\overset{.}{\varepsilon}}_{m}} \right)}} \right\rbrack^{2/3}},$

where

${{\overset{.}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}},$

the damage ω is called a test damage;

a numerical optimization algorithm is used to carry out a least squareoptimization on the analytical damage and the test damage, and thecorresponding constant q value is obtained.

In the S8, a fourth-order Runge-Kutta method is adopted to solve thestrain rate formula to obtain an evolution behavior of strain anddeformation with time; for the damage rate formula, a damage evolutionbehavior with time is obtained by using the formula

$\omega = {{- \frac{1}{q}}{{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}.}}$

Compared with the prior art, the technical scheme has the followingtechnical effects.

Firstly, the method for predicting creep damage and deformationevolution behavior with time provided by the application may accuratelypredict the minimum creep rate and creep life only by performinghigh-temperature tensile tests and high-temperature creep tests, and hasthe advantages of few required parameters, simple test, low cost andhigh precision;

Secondly, the method for predicting creep damage and deformationevolution behavior with time provided by the application is based on thecontinuous damage mechanics framework, and may continuously predict thecreep damage and deformation evolution with time, quantify the creepdamage, and when the creep time is 0, the damage is 0, and when thecreep time reaches the creep life, the damage is 1;

Thirdly, the method for predicting creep damage and deformationevolution behavior with time provided by the application takes intoaccount the uncertainty of creep data, which comes from many factors,including material dispersion, sample surface roughness, test deviation.Therefore, this method pays more attention to the average creep behaviorunder specific creep conditions, which represents the mid-value underthis condition, rather than the single creep curve behavior. Moreover,all the parameters in this method have clear stress-temperaturecorrelation, which makes this method have stronger interpolation andextrapolation capabilities; and

Fourthly, the method for predicting creep damage and deformationevolution behavior with time provided by the application may be appliedto alloy and other materials widely used in engineering, and has goodapplicability.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of threshold stress calculation method.

FIG. 2 is the reciprocal relationship between logarithmic minimum creeprate and temperature.

FIG. 3 is a diagram of linear fitting test data to solve (a) constantsn₁ and A₁ and (b) constants n₂ and A₂.

FIG. 4 is a graph showing the relationship between λ and stress andtemperature.

FIG. 5(A)-FIG. 5(B) is the evolution behavior of creep (a) strain and(b) damage with time at 600° C.

FIG. 6(A)-FIG. 6(B) is the evolution behavior of creep (a) strain and(b) damage with time at 650° C.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The technical scheme of the present application will be furtherexplained in detail with reference to the accompanying drawings.

The application discloses a method for predicting creep damage anddeformation evolution behavior with time, including:

S1, firstly, carrying out high-temperature tensile tests of materials atdifferent temperatures T to obtain the tensile strength σ_(b) atcorresponding temperatures;

S2, carrying out multiple groups of high-temperature creep tests underdifferent stress conditions at different temperatures to obtaincorresponding creep strain curves, minimum creep rate {dot over (ε)}_(m)and creep life t_(ƒ);

S3, establishing the relationship between the minimum creep rate {dotover (ε)}_(m), the stress σ and the threshold σ_(th) stress at the sametemperature by using the formula {dot over (ε)}_(m)=A_(m)(σ−σ_(th))⁵according to the minimum creep rate {dot over (ε)}_(m) data obtainedfrom the high-temperature creep test, where A_(m) is a constant; takingboth sides of formula {dot over (ε)}_(m)=A_(m)(σ−σ_(th))⁵ to the powerof ⅕ at the same time, and obtaining ({dot over (ε)}_(m))^(1/5)=A_(m)^(1/5) (σ−σ_(th)), where ({dot over (ε)}_(m))^(1/5) is the ordinate andσ is the abscissa, the test data at the same temperature are linearlyfitted, and the intercept between the fitted straight line and the Xaxis is the threshold stress σ_(th) at this temperature; carrying outthe same operation for different temperatures, and then get thethreshold stress σ_(th) corresponding to different temperatures;

S4, according to the corresponding tensile strength and threshold stressvalues at different temperatures obtained in S1 and S3, carrying outfitting by using polynomials, so as to establish the functionalrelationship between the tensile strength σ_(b) and the threshold stressσ_(th) and the temperature

$T,{\sigma_{b} = {\sum\limits_{i = 0}^{n}{a_{i}\left( T^{i} \right)}}},{\sigma_{th} = {\sum\limits_{i = 0}^{n}{b_{i}\left( T^{i} \right)}}},$

where n is the number of polynomial terms, a_(i) and b_(i) are fittingparameters, and i=0,1,2, . . . , n, n≤3.

S5, establishing the minimum creep rate {dot over (ε)}_(m) and the creeplife t_(ƒ) prediction formulas based on the threshold stress and thetensile strength respectively based on the threshold stress σ_(th) andthe tensile strength σ_(b) at a specific temperature obtained in theabove steps

${{\overset{.}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}}{{t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}}},}$

where A₁, A₂, n₁ and n₂ are constants, which may be obtained by linearfitting. σ_(th) is the threshold stress, σ_(b) is the tensile strength,σ is the stress, T is the temperature, and the unit is Kelvintemperature K, R is the gas constant (R=8.314J/(mol□K)). Q*_(N) is theapparent activation energy, which may be determined by the relationshipbetween the logarithm of the minimum creep rate and the reciprocal ofthe temperature under the same

$\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$

value. The specific method is as follows: when the

$\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$

values are the same, with {dot over (ε)}_(m) as the ordinate and thereciprocal 1/T of temperature as the abscissa, the experimental data arelinearly fitted, and the slope of the fitted straight line is

${- \frac{Q_{N}^{*}}{R}},$

and then the apparent activation energy Q*_(N) value is obtained.

The prediction formulas of minimum creep rate {dot over (ε)}_(m) andcreep life t_(ƒ) are mathematically transformed, and the logarithm ofboth sides of the equation is obtained.

${{\ln\left\lbrack {{\overset{.}{\varepsilon}}_{m}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack} = {{\frac{1}{n_{1}}{\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}} + {\frac{1}{n_{1}}{\ln\left( \frac{1}{A_{1}} \right)}}}}{{{\ln\left\lbrack {t_{f}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}} \right\rbrack} = {{\frac{1}{n_{2}}{\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}} + {\frac{1}{n_{2}}{\ln\left( \frac{1}{A_{2}} \right)}}}},}$

constants A₁ and n₁ may be obtained by the slope and intercept of thebest linear fitting line of

${\ln\left\lbrack {{\overset{.}{\varepsilon}}_{m}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack} - {\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}$

test data, respectively. Likewise constants A₂ and n₂ may be obtained bythe slope and intercept of the best linear fitting straight line of

${\ln\left\lbrack {t_{f}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}} \right\rbrack} - {\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}$

test data, respectively.

In this way, the minimum creep rate {dot over (ε)}_(m) and the creeplife t_(ƒ) at any stress temperature may be accurately predicted by theabove minimum creep rate {dot over (ε)}_(m) and the creep life t_(ƒ)prediction formulas. At a certain temperature, when the stressapproaches the threshold stress σ_(th), the minimum creep rate {dot over(ε)}_(m) tends to zero and the creep life tends to infinity; when thestress approaches the tensile strength σ_(b), the minimum creep rate{dot over (ε)}_(m) tends to infinity and the creep life tends to zero.

S6, establishing a creep damage constitutive model based on the minimumcreep rate {dot over (ε)}_(m) and creep life t_(ƒ) prediction formulasestablished in the S5:

${\overset{.}{\varepsilon} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}{\exp\left( {\lambda\omega}^{3/2} \right)}}}{{\overset{.}{\omega} = {{\left( \frac{1 - e^{q}}{q} \right)\left\lbrack {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack}^{- 1}{\exp\left( {q\omega} \right)}}},}$

where {dot over (ε)} is the strain rate, {dot over (ω)} is the damagerate, ε is the strain, ω is the damage, q is the constant related totemperature, and λ is the constant related to temperature and stress. Toensure that when the creep time reaches the creep life, when the creepfracture occurs, the damage is 1, and λ is defined as the logarithm ofthe ratio of the creep rate to the minimum creep rate at fracture,λ=1n(({dot over (ε)}_(final)/{dot over (ε)}_(m)). Using the λ valueobtained from the high-temperature creep test carried out in the S2, thedependence relationship between λ value and temperature stress may beestablished by linear fitting method, λ=(α₁T+α₂)σ+(α₃T+α₄), constantsα₁, α₂, α₃ and α₄ may be obtained by fitting the test data of λ withstress and temperature.

S7, integrating the damage rate formula in the S6,

${\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}},$

where

$t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{{\exp\left( {Q_{N}^{*}/{RT}} \right)}.}}$

The above-mentioned damage ω obtained by integration is calledanalytical damage.

Mathematically transform the strain rate formula in step 6 as follows:

${\omega = \left\lbrack {\frac{1}{\lambda}{\ln\left( {\overset{˙}{\varepsilon}/{\overset{˙}{\varepsilon}}_{m}} \right)}} \right\rbrack^{2/3}},$

where

${{\overset{˙}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}},$

this damage ω is called test damage.

At the same temperature, a numerical optimization algorithm is used tocarry out a least square optimization on the analytical damage and thetest damage, and the corresponding constant q value is obtained. Then,the functional relationship between the constant q value and thetemperature may be established by linearly fitting the temperature andthe constant q value, and the constant b₁ number and the constant b₂value in q=b₁T+b₂ may be obtained. Through the above S5-7, all theparameters in the creep damage constitutive model are uniquelydetermined.

S8, after all the parameters in the damage constitutive model aredetermined by the above steps, the fourth-order Runge-Kutta method isadopted to solve the strain, and the evolution behavior of strain anddeformation with time may be obtained. Specifically, the analyticaldamage ω obtained by integration is brought into the strain rateformula, and the creep rate {dot over (ε)}_(m) corresponding to anymoment t_(n) may be obtained. For the strain ε_(n) of at any time t_(n),the fourth-order Runge-Kutta algorithm may be used to calculate thestrain increment of each adjacent time interval, and the strain ε_(n)may be solved by the method of accumulation,

${\varepsilon_{n} = {\varepsilon_{0} + {\frac{1}{6}{\sum\limits_{i = 1}^{n}\left\{ {\left\lbrack {{\overset{˙}{\varepsilon}\left( t_{i - 1} \right)} + {\overset{˙}{\varepsilon}\left( t_{i} \right)} + {4{\overset{˙}{\varepsilon}\left( \frac{t_{i - 1} + t_{i}}{2} \right)}}} \right\rbrack*\left( {t_{i}\  - t_{i - 1}} \right)} \right\}}}}},$

where ε₀=0=, t₀=0.

For the damage, the analytical damage formula

$\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}$

may be used to obtain the evolution behavior of damage with time. Whent=t_(ƒ), the creep fracture occurs, the damage ω=1. In this way, thecreep damage and deformation evolution with time may be described.

In the application, the high-temperature tensile test of the materialaims at obtaining the corresponding tensile strength σ_(b) of thematerial at different temperatures T, and providing necessary parameterinput for subsequent high-temperature creep test, minimum creep rate andcreep life prediction method based on threshold stress and tensilestrength, and determination of creep damage constitutive model.

High-temperature creep test of materials are as follows: creep testsunder multiple groups of stresses are conducted at differenttemperatures, generally 2-4 temperature values may be selected, and 5-7groups of high-temperature creep tests under different stresses may beconducted at each temperature value. Until the creep fracture of thematerial occurs, the corresponding creep strain curves, minimum creeprate {dot over (ε)}_(m) and creep life t_(ƒ) under different stress andtemperature conditions are obtained.

The test instruments adopted by the application include anelectro-hydraulic servo fatigue tester and a creep tester.

The application will be further explained with reference to thefollowing specific embodiments.

Embodiment

In this embodiment, the method for predicting creep damage anddeformation evolution behavior with time of the present application isapplied to the prediction of creep damage and deformation of nickel-basesuperalloy GH4169, including the following steps.

(1) The high-temperature tensile test of GH4169 material is carried outat 600° C. and 650° C., and the corresponding tensile strength is 1440MPa and 1255 MPa, respectively.

(2) The high-temperature creep tests of GH4169 material with sixdifferent stress values are carried out at 600° C. and 650° C.respectively, and the corresponding creep strain curves, minimum creeprate {dot over (ε)}_(m) and creep life t_(ƒ) are obtained. The specifictest scheme and the obtained test data are shown in Table 1.

TABLE 1 creep test scheme and data of gh4169 material Minimum CreepTemperature Stress creep life No. (° C.) (MPa) rate(/h) (h) 1 600 9250.000123 92.23 2 880 0.000032 246 3 850 0.000022 326 4 820 0.000016416.33 5 805 0.000014 478 6 790 0.0000066 905 7 650 820 0.00059 17 8 7700.00013 72 9 720 0.00011 99 10 670 0.00007 139 11 615 0.000041 195 12595 0.000025 328.5

(3) Using the formula {dot over (ε)}_(m)=A_(m)(σ−σ_(th))⁵, the ({dotover (ε)}_(m))^(1/5)—σ data are linearly fitted at 600° C. and 650° C.respectively, and the stress value corresponding to the intersection ofthe fitted straight line and the X axis is the threshold stress at thistemperature. The calculated threshold stress is shown in FIG. 1 , andthe threshold stress is 593 MPa at 600° C. and 309 MPa at 650° C. Usingthe threshold stresses at these two temperatures, the threshold stressesat other temperatures may be calculated by linear interpolation orextrapolation.

(4) Based on the above obtained tensile strength σ_(b) at 600° C. and650° C. and threshold stress level σ_(th), polynomial fitting may beused to establish the functional relationship between tensile strengthand threshold stress and temperature respectively. Because theexperiment only carried out two temperatures, the linear fitting methodis adopted, and the first two terms in polynomial form are taken. Thefunctional relations between tensile strength and threshold stress andtemperature are obtained as follows: σ_(b)=−3.7*T+4670.1,σ_(th)=−5.68*T+5551.64 where T is Kelvin temperature.

(5) Based on the threshold stress σ_(th) and tensile strength σ_(b) at600° C. and 650° C. obtained by the above steps, the minimum creep rate{dot over (ε)}_(m) and creep life t_(ƒ) prediction formulas based on thethreshold stress and tensile strength are established respectively:

${\overset{˙}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/_{n_{1}}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}$${t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}}},$

firstly, under the same

$\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$

value, the Q*_(N) value of the apparent activation energy is determinedby the linear fitting relationship between the logarithm of the minimumcreep rate and the reciprocal of the temperature. Under the same

$\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$

value, the 1n

${\overset{˙}{\varepsilon}}_{m} - \frac{1}{T}$

test data is linearly fitted, and the slope is

${- \frac{Q_{N}^{*}}{R}},$

and then Q*_(N)=17128J/mol is obtained, as shown in FIG. 2 .

Then, the minimum creep rate {dot over (ε)}_(m) and creep life t_(ƒ)prediction formulas are mathematically transformed, and the logarithm ofboth sides of the equation is taken at the same time to obtain:

${\ln\left\lbrack {{\overset{˙}{\varepsilon}}_{m}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack} = {{\frac{1}{n_{1}}{\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}} + {\frac{1}{n_{1}}{\ln\left( \frac{1}{A_{1}} \right)}}}$${{\ln\left\lbrack {t_{f}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}} \right\rbrack} = {{\frac{1}{n_{2}}{\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}} + {\frac{1}{n_{2}}{\ln\left( \frac{1}{A_{2}} \right)}}}},$

The unknown parameters A₁ and n₁ may be obtained by linearly fitting theexperimental data of

${{\ln\left\lbrack {{\overset{˙}{\varepsilon}}_{m}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack} - {\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}},$

and The values of unknown parameters A₁ and n_(i) can be obtained byusing the slope and intercept of the corresponding fitting line.Similarly, by linearly fitting

${\ln\left\lbrack {t_{f}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}} \right\rbrack} - {\ln\left( \frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma} \right)}$

experimental data, the values of unknown parameters A₂ and n₂ may beobtained by using the slope and intercept of the corresponding fittingline. The fitted straight line is shown in FIG. 3 , and thedetermination coefficients of the fitted straight line are 0.9377 and0.9296 respectively, obtaining A₁=8.6249, n₁=0.3602, A₂=1.8529,n₂=−0.4244 by fitting. Therefore, the prediction formulas of minimumcreep rate {dot over (ε)}_(m) and creep life t_(ƒ) are obtained:

${\overset{˙}{\varepsilon}}_{m} = {\left( {\frac{1}{8.6249}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/0.3602}{\exp\left( {{- 1}7128/\left( {8.314T} \right)} \right)}}$$t_{f} = {\left( {\frac{1}{1.8529}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{{- 1}/0.4244}{{\exp\left( {1712{8/\left( {{8.3}14T} \right)}} \right)}.}}$

(6) Based on the prediction formula of the minimum creep rate {dot over(ε)}_(m) and creep life t_(ƒ), the creep damage constitutive model isestablished:

$\overset{˙}{\varepsilon} = {\left( {\frac{1}{{8.6}249}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/0.3602}{\exp\left( {{- 1}7128/\left( {8.314T} \right)} \right)}{\exp\left( {\lambda\omega^{3/2}} \right)}}$${\overset{.}{\omega} = {{\left( \frac{1 - e^{- q}}{q} \right)\left\lbrack {\left( {\frac{1}{1.8529}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{{- 1}/0.4244}{\exp\left( {1712{8/\left( {{8.3}14T} \right)}} \right)}} \right\rbrack}^{- 1}{\exp\left( {q\omega} \right)}}},$

where {dot over (ε)} is the strain rate, {dot over (ω)} is the damagerate, ε is the strain, ω is the damage, q is the constant related totemperature, λ is the constant related to temperature and stress. λ isdefined as the logarithm of the ratio of creep rate to minimum creeprate at fracture, λ=1n(({dot over (ε)}_(final)/{dot over (ε)}_(m)).According to the high-temperature creep test data, the λ valuecorresponding to the high-temperature creep test is shown in FIG. 4 :

The fitting formula λ=(α₁T+α₂)σ+(α₃T+α₄) is used to fit the λ testresults, and α₁=1.76*10-4, α₂=−0.180, α₃=−0.198 and α₄=202.6 areobtained. Therefore, λ=(1.76*10⁻⁴ T−0.180)σ+(−0.198T+202.6) is obtained.

(7) Integrating the damage rate formula in step (6):

${\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}},$

where,

$t_{f} = {\left( {\frac{1}{1.8529}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{{- 1}/0.4244}{{\exp\left( {17128/\left( 8.3147^{\prime} \right)} \right)}.}}$

The above-mentioned damage ω obtained by integration is calledanalytical damage.

Mathematically transform the strain rate formula in step (6):

${\omega = \left\lbrack {\frac{1}{\lambda}{\ln\left( {\overset{.}{\varepsilon}/{\overset{.}{\varepsilon}}_{m}} \right)}} \right\rbrack^{2/3}},$

where

${{\overset{.}{\varepsilon}}_{m} = {\left( {\frac{1}{8.6249}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/0.3602}{\exp\left( {{- 17128}/\left( 8.3147^{\prime} \right)} \right)}}},$

the damage ω is called test damage.

At the same temperature, the numerical optimization algorithm is used tocarry out a least square optimization on the analytical damage and thetest damage, and the corresponding constant q value is obtained. The qvalue is 2.4652 at 600° C. and 3.4842 at 650° C. Then, by linearlyfitting the temperature and the constant q value, the functionalrelationship between the constant q value and the temperature isestablished, and the constants b₁=0.0204 and b₂=−15.3265 in q=b₁T+b₂ areobtained. Then the expression of the constant q value isq=0.0204T-15.3265.

(8) After all the parameters in the damage constitutive model aredetermined through the above steps, the fourth-order Runge-Kutta methodis used to solve the strain, and the evolution behavior of strain anddeformation with time may be obtained. Specifically, the analyticaldamage ω obtained by integration is brought into the strain rateformula, and the creep rate {dot over (ε)}_(m) corresponding to at anytime t_(n) may be obtained. For the strain ε_(n) of at any time t_(n),the fourth-order Runge-Kutta algorithm may be used to calculate thestrain increment of each adjacent time interval, and the strain ε_(n)may be solved by the method of accumulation:

${\varepsilon_{n} = {\varepsilon_{0} + {\frac{1}{6}{\sum\limits_{i = 1}^{n}\left\{ {\left\lbrack {{\overset{.}{\varepsilon}\left( t_{i - 1} \right)} + {\overset{.}{\varepsilon}\left( t_{i} \right)} + {4{\overset{.}{\varepsilon}\left( \frac{t_{i - 1} + t_{i}}{2} \right)}}} \right\rbrack*\left( {t_{i} - t_{i - 1}} \right)} \right\}}}}},$

where ε₀=0, t₀=0

For damage, the formula

$\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}$

may be used to obtain the evolution behavior of damage with time. Whent=t_(ƒ), the creep fracture occurs, the damage ω=1. In this way, thecreep damage and deformation evolution with time may be predicted. Theevolution behaviors of creep strain and damage with time at 600° C. and650° C. are shown in FIG. 5(A)-FIG. 5(B) and FIG. 6(A)-FIG. 6(B),respectively.

Therefore, the prediction of creep damage and deformation evolution withtime under arbitrary temperature stress may be solved by the creepdamage constitutive equation combined with the least square optimizationalgorithm and the fourth-order Runge-Kutta algorithm. The damageconstitutive model formula is as follows:

$\overset{.}{\varepsilon} = {\left( {\frac{1}{8.6249}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/0.3602}{\exp\left( {{- 17128}/\left( 8.3147^{\prime} \right)} \right)}{\exp\left( {\lambda\omega}^{3/2} \right)}}$${\overset{.}{\omega} = {{\left( \frac{1 - e^{- q}}{q} \right)\left\lbrack {\left( {\frac{1}{1.8529}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{{- 1}/0.4244}{\exp\left( {17128/\left( 8.3147^{\prime} \right)} \right)}} \right\rbrack}^{- 1}{\exp\left( {q\omega} \right)}}},$

where λ=(1.76*10⁻⁴T−0.180)σ+(−0.198T+202.6), q=0.0204T−15.3265σ_(b)=−3.7*T+4670.1, σ_(th)=−5.68*T+5551.64. To sum up, thestress-temperature correlations of all parameters in the model areclearly characterized, which makes the method applicable to any stressand temperature conditions and has strong extrapolation ability.

It may also be seen from FIG. 5(A)-FIG. 5(B) and FIG. 6(A)-FIG. 6(B)that this method models the average creep behavior under the sameconditions, rather than a single creep curve. This method represents themedian situation under this condition, and the predictions of creepdamage and deformation almost all fall in the 20% life dispersion zone.The predicted results are in good agreement with the experimentalresults, showing satisfactory prediction accuracy, and reliableinterpolation and extrapolation may be realized.

The above are only the preferred embodiments of the present application,and it should be pointed out that for those of ordinary skill in thetechnical field, without departing from the principle of the presentapplication, several improvements and modifications may be made, andthese improvements and modifications should fall in the protection scopeof the present application.

What is claimed is:
 1. A method for predicting creep damage and deformation evolution behaviors with time, comprising: S1, carrying out high-temperature tensile tests of materials at different temperatures T to obtain tensile strength σ_(b) at different temperatures; S2, carrying out high-temperature creep tests under different stress conditions at different temperatures to obtain corresponding creep strain curves, a minimum creep rate {dot over (ε)}_(m) and a creep life t_(ƒ); S3, obtaining threshold stresses σ_(th) corresponding to different temperatures according to the minimum creep rate {dot over (ε)}_(m) obtained in the S2; S4, establishing a functional relationship between the tensile strength σ_(b), the threshold stress σ_(th) and the temperature T according to the tensile strength σ_(b) at different temperatures obtained in the S1 and the threshold stresses σ_(th) at different temperatures obtained in the S3; S5, establishing prediction formulas of the minimum creep rate {dot over (ε)}_(m) and the creep life t_(ƒ) based on the threshold stress σ_(th) obtained in the S3 and the tensile strength σ_(b), obtained in the S1 respectively, and predicting a minimum creep rate {dot over (ε)}_(m) and a creep life t_(ƒ) under any stress temperature conditions with the prediction formulas; S6, establishing a creep damage constitutive model based on the prediction formulas of the minimum creep rate {dot over (ε)}_(m) and the creep life t_(ƒ) established in the S5, wherein the creep damage constitutive model comprises a strain rate formula and a damage rate formula; S7, determining parameters in the creep damage constitutive model established in the S6; and S8, obtaining an evolution behavior of strain deformation with time by solving the strain rate formula; and obtaining an evolution behavior of damage with time by solving the damage rate formula.
 2. The method for predicting creep damage and deformation evolution behaviors with time according to claim 1, wherein in the S3, a relationship between the minimum creep rate {dot over (ε)}_(m), the stress σ and the threshold stress σ_(th) at a same temperature is established by using the formula {dot over (ε)}_(m)=A_(m)(σ−σ_(th))⁵ according to the minimum creep rate {dot over (ε)}_(m) data obtained from the high-temperature creep tests in the S2, A_(m) is a constant, a same operation for different temperatures is carried out, and then threshold stress levels corresponding to different temperatures are obtained.
 3. The method for predicting creep damage and deformation evolution behaviors with time according to claim 1, wherein in the S4, a functional relationship between the tensile strength σ_(b), the threshold stress σ_(th) and the temperature T is established according to the tensile strength σ_(b) at different temperatures obtained in the S1 and the threshold stresses σ_(th) at different temperatures obtained in the S3, and a polynomial is used for fitting, ${\sigma_{b} = {\sum\limits_{i = 0}^{n}{a_{i}(T)}^{i}}},$ $\sigma_{th} = {\sum\limits_{i = 0}^{n}{b_{i}(T)}^{i}}$ with n as a number of polynomial terms, a₁, b₁ as fitting parameters, and i=0,1,2 . . . ,n, n≤3.
 4. The method for predicting creep damage and deformation evolution behaviors with time according to claim 1, wherein in the S5, the prediction formulas of the minimum creep rate {dot over (ε)}_(m) and the creep life t_(ƒ) based on the threshold stress σ_(th) and the tensile strength σ_(b) are respectively established based on the threshold stress σ_(th) obtained in the S3 and the tensile strength σ_(b) obtained in the S1: ${\overset{.}{\varepsilon}}_{m} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}}$ ${t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}}},$ wherein A₁, A₂, n₁ and n₂ are constants, σ_(th) is the threshold stress, σ_(b) is the tensile strength, σ is applied stress, T is applied temperature, R is a gas constant, and Q*_(N) is apparent activation energy; and the minimum creep rate {dot over (ε)}_(m) and creep life t_(ƒ) may be predicted with the above two expressions under arbitrary stress temperature conditions.
 5. The method for predicting creep damage and deformation evolution behaviors with time according to claim 4, wherein in the S5, the apparent activation energy Q*_(N) is obtained in a following method: under a same $\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}$ value, the apparent activation energy Q*_(N) is determined by a linear fitting straight line slope between a logarithm 1n {dot over (ε)}_(m) of the minimum creep rate and the reciprocal 1/T of the temperature.
 6. The method for predicting creep damage and deformation evolution behaviors with time according to claim 4, wherein in the S6, a creep damage constitutive model is established based on the prediction formulas of the minimum creep rate {dot over (ε)}_(m) and the creep life t_(ƒ) in the S5: $\overset{.}{\varepsilon} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{N}^{*}}/{RT}} \right)}{\exp\left( {\lambda\omega}^{3/2} \right)}}$ ${\overset{.}{\omega} = {{\left( \frac{1 - e^{- q}}{q} \right)\left\lbrack {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{{- 1}/n_{2}}{\exp\left( {Q_{N}^{*}/{RT}} \right)}} \right\rbrack}^{- 1}{\exp\left( {q\omega} \right)}}},$ wherein {dot over (ε)} is the strain rate, {dot over (ω)} is the damage rate, ε is the strain and ω is the damage, q is a constant related to a temperature, λ is a constant related to the temperature and the stress; in order to ensure that when creep fracture occurs, the damage is 1, λ is defined as a logarithm of the creep rate {dot over (ε)}_(final) to the minimum creep rate {dot over (ε)}_(m) when creep fracture occurs, λ=1n({dot over (ε)}_(final)/{dot over (ε)}_(m)); fitting the experimental data, and the expression of λ is established as λ=(α₁T+α₂)σ+(α₃T+α₄), and α₁, α₂, α₃ and α₄ are fitting parameters.
 7. The method for predicting creep damage and deformation evolution behaviors with time according to claim 6, wherein in the S7, the damage rate formula in the S6 is integrated, obtaining: ${\omega = {{- \frac{1}{q}}{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}}},$ wherein ${t_{f} = {\left( {\frac{1}{A_{2}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{2}}{\exp\left( {Q_{n}^{*}/{RT}} \right)}}},$ the damage ω obtained by an integral is called an analytical damage; the strain rate formula is mathematically transformed in the S6 as follows: ${\omega = \left\lbrack {\frac{1}{\lambda}{\ln\left( {\overset{.}{\varepsilon}/{\overset{.}{\varepsilon}}_{m}} \right)}} \right\rbrack^{2/3}},{t_{f} = {\left( {\frac{1}{A_{1}}\frac{\sigma - \sigma_{th}}{\sigma_{b} - \sigma}} \right)^{1/n_{1}}{\exp\left( {{- Q_{n}^{*}}/{RT}} \right)}}},$ a damage ω is called a test damage; and a numerical optimization algorithm is used to carry out a least square optimization on the analytical damage and the test damage, and a corresponding constant q value is obtained.
 8. The method for predicting creep damage and deformation evolution behavior with time according to claim 7, wherein in the S8, a fourth-order Runge-Kutta method is adopted to solve the strain rate formula to obtain an evolution behavior of strain and deformation with time; for the damage rate formula, a damage evolution behavior with time is obtained by using the formula $\omega = {{- \frac{1}{q}}{{\ln\left\lbrack {1 - {\left( {1 - e^{- q}} \right)\frac{t}{t_{f}}}} \right\rbrack}.}}$ 